Applied Mathematics

Interests

My research interest lies in the theory and applications of probabilistic methods. Recently, I have been working on game-theoretic importance sampling (IS). Importance sampling is a variance reduction technique in Monte Carlo simulation, and can be especially effective when the quantities of interest are largely determined by rare events.

I have always been interested in the theory and applications of probabilistic methods. My previous work was largely on mathematical finance, where many fundamental questions, such as option pricing and portfolio optimization, can be answered using the technical tools from stochastic optimization theory. My main contribution is the study of explicit solution to path-dependent options with a double exponential jump diffusion stock price, and the convex duality method for portfolio optimization in an incomplete semi-martingale financial market.

Recently, I have been working on game-theoretic importance sampling (IS). Importance sampling is a variance reduction technique in Monte Carlo simulation, and can be especially effective when the quantities of interest are largely determined by rare events. Its basic idea is to sample the system under a different probability distribution, and the efficiency of the algorithm depends on the choice of this alternative sampling distribution, which is the central question of importance sampling. In close collaboration with Paul Dupuis, we introduced a game theoretic approach toward importance sampling, and showed that IS algorithms are intimately connected with a differential game. The corresponding Isaacs equation and its subsolutions can serve as a very useful tool for the design and analysis of IS algorithms. The significance of this game formulation is that, for the first time, it allows one to build a comprehensive framework within which one can systematically and rigorously construct efficient dynamic IS schemes.

Much still remains to be done in the theory and applications of game-theoretic importance sampling. For example, one can consider the extension of importance sampling to different settings such as Jackson queuing networks, networks with Markov modulated arrival and service rates, and functionals of exit time and location for a small noise process in a region surrounding a stable rest point. Another interesting area I hope to investigate is the robust control for general queuing networks. There seem to be very few concrete, explicit results, especially in high dimensions. This difficulty is largely due to the dimensionality and the nature of constrained dynamics associated with such systems. I am interested in developing network models and cost structures such that a fairly complete characterization of the optimal policies are possible, regardless of dimension.

Affiliations

Refereeing papers for journals including: Annals of Probability, Annals of Applied Probability, Applied Probability, Applied Mathematics and Optimization, Mathematics of Operations research, SIAM Journal on Control and Optimization, Mathematical Finance, Finance and Stochastics, Stochastic Analysis and Its Applications, Stochastics and Stochastics Reports.

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